Olympiad problems

1985 Czech And Slovak Olympiad IIIA, 1

A regular $1985$-gon is given in the plane. Let's pass a straight line through each side of it. Determine the number of parts into which these lines divide the plane.

2016 Sharygin Geometry Olympiad, 4

Tags: polygon , cut , regular polygon , geometry

Is it possible to dissect a regular decagon along some of its diagonals so that the resulting parts can form two regular polygons? by N.Beluhov

2013 Korea Junior Math Olympiad, 8

Tags: combinatorial geometry , regular polygon , combinatorics

Drawing all diagonals in a regular $2013$-gon, the regular $2013$-gon is divided into non-overlapping polygons. Prove that there exist exactly one $2013$-gon out of all such polygons.

2023 Israel TST, P1

Tags: regular polygon , combinatorics

A regular polygon with $20$ vertices is given. Alice colors each vertex in one of two colors. Bob then draws a diagonal connecting two opposite vertices. Now Bob draws perpendicular segments to this diagonal, each segment having vertices of the same color as endpoints. He gets a fish from Alice for each such segment he draws. How many fish can Bob guarantee getting, no matter Alice's goodwill?

2000 Tuymaada Olympiad, 7

Tags: combinatorics , combinatorial geometry , geometry , regular polygon , pentagon

Every two of five regular pentagons on the plane have a common point. Is it true that some of these pentagons have a common point?

1984 Brazil National Olympiad, 3

Tags: geometry , rectangle , dodecahedron , regular polygon , 3d geometry

Given a regular dodecahedron of side $a$. Take two pairs of opposite faces: $E, E' $ and $F, F'$. For the pair $E, E'$ take the line joining the centers of the faces and take points $A$ and $C$ on the line each a distance $m$ outside one of the faces. Similarly, take $B$ and $D$ on the line joining the centers of $F, F'$ each a distance $m$ outside one of the faces. Show that $ABCD$ is a rectangle and find the ratio of its side lengths.